2 marksStatisticsRandom variables and standard distributionsThe normal distributionHardabout 3 min

IIT JAM 2021 Question 57: Random variables and standard distributions

Let $X_1, X_2, X_3, X_4$ be independent standard normal variables and $Y = (X_1 + X_2)^2 + (X_3 + X_4)^2$ . The variance of $Y$ equals _______ (in integer).

Answer and worked solution

Answer: 16

$X_1 + X_2 \sim N(0, 2)$ , so $(X_1 + X_2)^2 = 2\chi^2_1$ with variance $4 \cdot 2 = 8$ ; the same holds for $(X_3 + X_4)^2$ .

The two squared terms are independent, so $\text{Var}(Y) = 8 + 8 = 16$ . The answer is $16$ .

Plain-text version

Let X₁, X₂, X₃, X₄ be independent standard normal variables and Y = (X₁ + X₂)² + (X₃ + X₄)². The variance of Y equals _ (in integer).

Also written as: Let X_1, X_2, X_3, X_4 be independent standard normal variables and Y = (X_1 + X_2)^2 + (X_3 + X_4)^2. The variance of Y equals _ (in integer).

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What this question tests

This question is on The normal distribution, part of Random variables and standard distributions in Statistics.

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